A new method of constructing arcs in projective space is given. It is a generalisation of the fac... more A new method of constructing arcs in projective space is given. It is a generalisation of the fact that a normal rational curve can be given by the complete intersection of a set of quadrics. The non-classicallO-arc of PG(4, 9) together with its special point is the set of derived points of a. cubic primal. This property is shared with the normal rational curve of this spaceo L INTRODUCTION AND NOTATION This paper shows that quadrics (quadratic or hypersurfaces in finite projective space) are related to k-arcs and their associated curves in a fundamental way. The methods of classical algebraic geometry are necessary for much of the discussion; see [1,26]. Before delving into these connections we shall give the reader a gentle introduction to the known constructions and theory of arcs. The geometries which are the object for this discussion are, in the main, the finite projective spaces PG(n,q) of dimension n over GF(q), for n 2: 2, and q = ph, P prime. However, many of the results a...
We discuss the recently introduced concept of k-in-out graphs, and provide a construction for k-i... more We discuss the recently introduced concept of k-in-out graphs, and provide a construction for k-in-out graphs for any positive integer k. We derive a lower bound for the number of vertices of a k-in-out graph for any positive integer k, and demonstrate that our construction meets this bound in all cases. For even k, we also prove our construction is optimal with respect to the number of edges, and results in a planar graph. Among the possible uses of in-out graphs, they can convert the generalized traveling salesman problem to the asymmetric traveling salesman problem, avoiding the “big M” issue present in most other conversions. We give constraints satisfied by all in-out graphs to assist cutting-plane algorithms in solving instances of traveling salesman problem which contain in-out graphs.
Users may download and print one copy of any publication from the public portal for the purpose... more Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.
In August 2015 NORDITA (Nordic Institute for Theoretical Physics) hosted a conference where Hawki... more In August 2015 NORDITA (Nordic Institute for Theoretical Physics) hosted a conference where Hawking strongly supported the conjectured relationship between string theory and quantum fields that was initiated with the holographic principle some 20 years ago by ’t Hooft, Maldacena, Susskind and Witten, and more recently van Raamsdonk. We bring together results of several papers showing how mathematics can come to the party: the fundamentals of flat (even higher-dimensional) space can be derived very simply from topological properties on a surface. Specifically, Desargues, Pappus and other configurations do not have to be assumed a priori or self-evident (a fundamental weakness of Hilbert's work 1899) to develop the foundations of geometry. Are black holes places where non-commutative (quantum) behavior reigns while Euclidean (flat) space is where commutativity holds sway? So we cannot hope to look inside a black hole unless we know how “deformable” topology is related to “flat” ge...
A note on Nk configurations and theorems in projective space
Bulletin of the Australian Mathematical Society, 2007
A method of embedding nk configurations into projective space of k–1 dimensions is given. It brea... more A method of embedding nk configurations into projective space of k–1 dimensions is given. It breaks into the easy problem of finding a rooted spanning tree of the associated Levi graph. Also it is shown how to obtain a “complementary” “theorem” about projective space (over a field or skew-field F) from any nk theorem over F. Some elementary matroid theory is used, but with an explanation suitable for most people. Various examples are mentioned, including the planar configurations: Fano 73, Pappus 93, Desargues 103 (also in 3d-space), Möbius 84 (in 3d-space), and the resulting 74 in 3d-space, 96 in 5d-space, and 107 in 6d-space. (The Möbius configuration is self-complementary.) There are some nk configurations that are not embeddable in certain projective spaces, and these will be taken to similarly not embeddable configurations by complementation. Finally, there is a list of open questions.
A geometrical representation theory for orthogonal arrays
Bulletin of the Australian Mathematical Society, 1994
Every orthogonal array of strength s and of prime-power (or perhaps infinite) order q, has a well... more Every orthogonal array of strength s and of prime-power (or perhaps infinite) order q, has a well-defined collection of ranks r. Having rank r means that it can be constructed as a cone cut by qs hyperplanes in projective space of dimension r over a field of order q.
The modular counterparts of Cayley's hyperdeterminants
Bulletin of the Australian Mathematical Society, 1998
Let H be a hypersurface of degree m in PG(n, q), q = ph, p prime.(1) If m < n + 1, H has 1 (mo... more Let H be a hypersurface of degree m in PG(n, q), q = ph, p prime.(1) If m < n + 1, H has 1 (mod p) points.(2) If m = n + 1, H has 1 (mod p) points ⇔ Hp−1 has no term We show some applications, including the generalised Hasse invariant for hypersurfaces of degree n + 1 in PG(n, F), various porperties of finite projective spaces, and in particular a p-modular invariant detp of any (n + 1)r+2 = (n + 1)×…×(n + 1) array on hypercube A over a field characteristic p. This invariant is multiplicative in that detp(AB) = detp(B), whenever the product (or convolution of the two arrays A and B is defined, and both arrays are not 1-dimensional vectors. (If A is (n + 1)r+2 and B is (n + 1)s+2, then AB is (n + 1)r+s+2.) The geometrical meaning of the invariant is that over finite fields of characteristic p the number of projections of A from r + 1 points in any given r + 1 directions of the array to a non-zero point in the final direction is 0 (mod p). Equivalently, the number of projections of...
Theorems of Points and Planes in Three-Dimensional Projective Space
Journal of the Australian Mathematical Society, 2010
We discuss n4 configurations of n points and n planes in three-dimensional projective space. Thes... more We discuss n4 configurations of n points and n planes in three-dimensional projective space. These have four points on each plane, and four planes through each point. When the last of the 4n incidences between points and planes happens as a consequence of the preceding 4n−1 the configuration is called a ‘theorem’. Using a graph-theoretic search algorithm we find that there are two 84 and one 94 ‘theorems’. One of these 84 ‘theorems’ was already found by Möbius in 1828, while the 94 ‘theorem’ is related to Desargues’ ten-point configuration. We prove these ‘theorems’ by various methods, and connect them with other questions, such as forbidden minors in graph theory, and sets of electrons that are energy minimal.
This paper studies families of self-orthogonal codes over Z 4 . We show that the simplex codes (o... more This paper studies families of self-orthogonal codes over Z 4 . We show that the simplex codes (of Type α and Type β) are self-orthogonal. We answer the question of Z 4 -linearity for some codes coming from projective planes of even order. A new family of self-orthogonal codes over Z 4 is constructed via projective planes of odd order.
A new method of constructing arcs in projective space is given. It is a generalisation of the fac... more A new method of constructing arcs in projective space is given. It is a generalisation of the fact that a normal rational curve can be given by the complete intersection of a set of quadrics. The non-classicallO-arc of PG(4, 9) together with its special point is the set of derived points of a. cubic primal. This property is shared with the normal rational curve of this spaceo L INTRODUCTION AND NOTATION This paper shows that quadrics (quadratic or hypersurfaces in finite projective space) are related to k-arcs and their associated curves in a fundamental way. The methods of classical algebraic geometry are necessary for much of the discussion; see [1,26]. Before delving into these connections we shall give the reader a gentle introduction to the known constructions and theory of arcs. The geometries which are the object for this discussion are, in the main, the finite projective spaces PG(n,q) of dimension n over GF(q), for n 2: 2, and q = ph, P prime. However, many of the results a...
We discuss the recently introduced concept of k-in-out graphs, and provide a construction for k-i... more We discuss the recently introduced concept of k-in-out graphs, and provide a construction for k-in-out graphs for any positive integer k. We derive a lower bound for the number of vertices of a k-in-out graph for any positive integer k, and demonstrate that our construction meets this bound in all cases. For even k, we also prove our construction is optimal with respect to the number of edges, and results in a planar graph. Among the possible uses of in-out graphs, they can convert the generalized traveling salesman problem to the asymmetric traveling salesman problem, avoiding the “big M” issue present in most other conversions. We give constraints satisfied by all in-out graphs to assist cutting-plane algorithms in solving instances of traveling salesman problem which contain in-out graphs.
Users may download and print one copy of any publication from the public portal for the purpose... more Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.
In August 2015 NORDITA (Nordic Institute for Theoretical Physics) hosted a conference where Hawki... more In August 2015 NORDITA (Nordic Institute for Theoretical Physics) hosted a conference where Hawking strongly supported the conjectured relationship between string theory and quantum fields that was initiated with the holographic principle some 20 years ago by ’t Hooft, Maldacena, Susskind and Witten, and more recently van Raamsdonk. We bring together results of several papers showing how mathematics can come to the party: the fundamentals of flat (even higher-dimensional) space can be derived very simply from topological properties on a surface. Specifically, Desargues, Pappus and other configurations do not have to be assumed a priori or self-evident (a fundamental weakness of Hilbert's work 1899) to develop the foundations of geometry. Are black holes places where non-commutative (quantum) behavior reigns while Euclidean (flat) space is where commutativity holds sway? So we cannot hope to look inside a black hole unless we know how “deformable” topology is related to “flat” ge...
A note on Nk configurations and theorems in projective space
Bulletin of the Australian Mathematical Society, 2007
A method of embedding nk configurations into projective space of k–1 dimensions is given. It brea... more A method of embedding nk configurations into projective space of k–1 dimensions is given. It breaks into the easy problem of finding a rooted spanning tree of the associated Levi graph. Also it is shown how to obtain a “complementary” “theorem” about projective space (over a field or skew-field F) from any nk theorem over F. Some elementary matroid theory is used, but with an explanation suitable for most people. Various examples are mentioned, including the planar configurations: Fano 73, Pappus 93, Desargues 103 (also in 3d-space), Möbius 84 (in 3d-space), and the resulting 74 in 3d-space, 96 in 5d-space, and 107 in 6d-space. (The Möbius configuration is self-complementary.) There are some nk configurations that are not embeddable in certain projective spaces, and these will be taken to similarly not embeddable configurations by complementation. Finally, there is a list of open questions.
A geometrical representation theory for orthogonal arrays
Bulletin of the Australian Mathematical Society, 1994
Every orthogonal array of strength s and of prime-power (or perhaps infinite) order q, has a well... more Every orthogonal array of strength s and of prime-power (or perhaps infinite) order q, has a well-defined collection of ranks r. Having rank r means that it can be constructed as a cone cut by qs hyperplanes in projective space of dimension r over a field of order q.
The modular counterparts of Cayley's hyperdeterminants
Bulletin of the Australian Mathematical Society, 1998
Let H be a hypersurface of degree m in PG(n, q), q = ph, p prime.(1) If m < n + 1, H has 1 (mo... more Let H be a hypersurface of degree m in PG(n, q), q = ph, p prime.(1) If m < n + 1, H has 1 (mod p) points.(2) If m = n + 1, H has 1 (mod p) points ⇔ Hp−1 has no term We show some applications, including the generalised Hasse invariant for hypersurfaces of degree n + 1 in PG(n, F), various porperties of finite projective spaces, and in particular a p-modular invariant detp of any (n + 1)r+2 = (n + 1)×…×(n + 1) array on hypercube A over a field characteristic p. This invariant is multiplicative in that detp(AB) = detp(B), whenever the product (or convolution of the two arrays A and B is defined, and both arrays are not 1-dimensional vectors. (If A is (n + 1)r+2 and B is (n + 1)s+2, then AB is (n + 1)r+s+2.) The geometrical meaning of the invariant is that over finite fields of characteristic p the number of projections of A from r + 1 points in any given r + 1 directions of the array to a non-zero point in the final direction is 0 (mod p). Equivalently, the number of projections of...
Theorems of Points and Planes in Three-Dimensional Projective Space
Journal of the Australian Mathematical Society, 2010
We discuss n4 configurations of n points and n planes in three-dimensional projective space. Thes... more We discuss n4 configurations of n points and n planes in three-dimensional projective space. These have four points on each plane, and four planes through each point. When the last of the 4n incidences between points and planes happens as a consequence of the preceding 4n−1 the configuration is called a ‘theorem’. Using a graph-theoretic search algorithm we find that there are two 84 and one 94 ‘theorems’. One of these 84 ‘theorems’ was already found by Möbius in 1828, while the 94 ‘theorem’ is related to Desargues’ ten-point configuration. We prove these ‘theorems’ by various methods, and connect them with other questions, such as forbidden minors in graph theory, and sets of electrons that are energy minimal.
This paper studies families of self-orthogonal codes over Z 4 . We show that the simplex codes (o... more This paper studies families of self-orthogonal codes over Z 4 . We show that the simplex codes (of Type α and Type β) are self-orthogonal. We answer the question of Z 4 -linearity for some codes coming from projective planes of even order. A new family of self-orthogonal codes over Z 4 is constructed via projective planes of odd order.
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